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A rectangular glass slab ABCD of refract...

A rectangular glass slab ABCD of refractive index `n_(1)` is immerged in water of refractive index `n_(2) (n_(1) gt n_(2))` . A ray of light is incident at the surface AB of the slab as shown . The maximum value of the angle of incidence `alpha_("max")` , such that the ray comes out only from the other surface CD , is given by :-

A

` sin^(-1) [ (n_(1))/( n_(2)) cos ( sin^(-1) ""(n_(2))/( n_(1)))]`

B

` sin^(-1) [ n_(1) cos ( sin^(-1) "" (1)/( n_(2))) ]`

C

` sin^(-1) ((n_(1))/( n_(2)))`

D

` sin^(-1) ((n_(2))/(n_(1)))`

Text Solution

Verified by Experts

The correct Answer is:
A
`i + r = 90^(@)`
`rArr r = 90 - i` . . . (i)
and Snell.s law
`n_(2) sin alpha _("max") = n_(1) sin r`
`sin alpha _("max") = (n_(1))/( n_(2)) sin r`
`alpha _("max") = sin ^(-1) [ (n_(1))/( n_(2)) sin r] ` . . . (ii)
Now, `(i) rArr r = 90^(@) - i_(c) ` [ critical angle ]
`rArr = 90 - sin^(-1) ((n_(2))/( n_(1)))`
on putting the value of r in equation (ii)
`:. a_("max") = sin ^(-1) [ (n_(1))/( n_(2)) sin (90 - sin ^(-1) ((n_(2))/(n_(1))))]`
`= sin ^(-1) [ (n_(1))/(n_(2)) cos (sin^(-1) "" (n_(2))/(n_(1)))]`
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